Error Analysis of Nonconstant Admittivity for MR ... - Semantic Scholar
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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 31, NO. 2, FEBRUARY 2012
Error Analysis of Nonconstant Admittivity for MR-Based Electric Property Imaging Jin Keun Seo, Min-Oh Kim, Joonsung Lee, Narae Choi, Eung Je Woo, Member, IEEE, Hyung Joong Kim, Oh In Kwon, and Dong-Hyun Kim*, Member, IEEE
Abstract—Magnetic resonance electrical property tomography (MREPT) is a new imaging modality to visualize a distribution inside the human body where of admittivity and denote electrical conductivity and permittivity, respectively. Using B1 maps acquired by an magnetic resonance imaging and at the scanner, it produces cross-sectional images of Larmor frequency. Since current MREPT methods rely on an assumption of a locally homogeneous admittivity, there occurs a reconstruction error where this assumption fails. Rigorously analyzing the reconstruction error in MREPT, we showed that the error is fundamental and may cause technical difficulties in interpreting MREPT images of a general inhomogeneous object. We performed numerical simulations and phantom experiments to quantitatively support the error analysis. We compared the MREPT image reconstruction problem with that of magnetic resonance electrical impedance tomography (MREIT) to highlight distinct features of both methods to probe the same object in terms of its high- and low-frequency conductivity distributions, respectively. MREPT images showed large errors along boundaries where admittivity values changed whereas MREIT images showed no such boundary effects. Noting that MREIT makes use of the term neglected in MREPT, a novel MREPT admittivity image reconstruction method is proposed to deal with the boundary effects, which requires further investigation on the complex directional derivative in the real Euclidian space 3 .
= +
Index Terms—B1 map, conductivity, magnetic resonance electrical impedance tomography (MREIT), magnetic resonance electrical property tomography (MREPT), permittivity.
I. INTRODUCTION
T
OMOGRAPHIC imaging of electrical conductivity and permittivity distributions inside the human body has been an important research topic since they provide diagnostic information related with physiological and pathological conditions Manuscript received May 20, 2011; revised July 26, 2011, August 23, 2011; accepted September 26, 2011. Date of publication October 10, 2011; date of current version February 03, 2012. The work of J. K. Seo was supported by the WCU program through MEST/NRF (R31-2008-000-10049-0). The work of E. J. Woo was supported by the NRF grant funded by the Korea government (MEST) (20100018275). The work of D. H. Kim was supported by the KOSEF grant funded by the Korea government (MEST) (2010-0016421). Asterisk indicates corresponding author. J. K. Seo is with the Department of Computational Science and Engineering, Yonsei University, Seoul 120-749, South Korea (e-mail: [email protected]). M.-O. Kim and J. Lee, N. Choi are with the Department of Electrical and Electronic Engineering, Yonsei University, 120-749 Seoul, South Korea (e-mail: [email protected]). E. J. Woo and H. J. Kim are with the Department of Biomedical Engineering, Kyung Hee University, 446–701 Yongin, Gyeonggi, South Korea. O. I. Kwon is with the Department of Mathematics, Konkuk University, Seoul 143-701, South Korea. *D.-H. Kim is with the Department of Electrical and Electronic Engineering, Yonsei University, Seoul 120-749, South Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TMI.2011.2171000
of tissues and organs [5]–[8]. Throughout this paper, the conand permittivity are assumed to be ductivity isotropic and depend on the angular frequency and the po. The admittivity is denoted as sition . The human body is assumed to be homogeneous in terms of the magnetic permeability with the value of . During the last three decades, there have been numerous studies on electrical impedance tomography (EIT), which aims to reconstruct cross-sectional images of an admittivity distribution inside the human body [3], [10], [15]. Using boundary current-voltage data sets subject to multiple externally injected currents, a solution of an ill-posed nonlinear inverse problem provides cross-sectional images of the admittivity distribution. The measured boundary data sets are, however, insensitive to a local perturbation of the admittivity away from measuring points whereas they are sensitive to errors in the boundary geometry and electrode positions. Static EIT imaging to recover absolute admittivity values has not been successful so far. Though linearized difference EIT imaging methods are finding some clinical applications, the spatial resolution is relatively poor compared with other imaging modalities. For static admittivity image reconstructions, magnetic resonance electrical impedance tomography (MREIT) [2], [25], [26] and magnetic resonance electrical property tomography (MREPT) [9], [11] have been proposed to use internal data of induced magnetic fields. In MREIT, a low-frequency current below a few kilohertz is injected to induce a magnetic field , whose -component is measured from MR phase images using a magnetic resonance imaging (MRI) scanner with its main field in the -direction. The current-injection MRI technique to measure the induced data in a form of an image was originally proposed for magnetic resonance current density imaging (MRCDI) [21], [22]. Since MREIT relies on the magnetic field induced by low-frequency current, it can produce images of the conductivity distribution only at the low frequency. For MREIT conductivity image reconstructions, one may adopt the harmonic algorithm [20], which utilizes two data sets of and subject to two linearly independent injection currents and , respectively. Since the induced magnetic fields of and are influenced by the conductivity , the harmonic algorithm recovers based on the relation between the induced magnetic field and the conductivity. Numerical simulations and imaging experiments of phantoms, animals, and human subjects showed that MREIT can produce low-frequency conductivity images with a pixel size of about one millimeter using imaging currents of a few milliamperes [12], [13], [16], [18], [19]. For in vivo
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SEO et al.: ERROR ANALYSIS OF NONCONSTANT ADMITTIVITY FOR MR-BASED ELECTRIC PROPERTY IMAGING
human imaging, the sensitivity of the MR-based measurement technique needs to be enhanced to reduce the amount of injection current down to a level where internal current density distributions do not induce nerve stimulation. maps [1], [4], [17], [23], [28] that are inMREPT utilizes fluenced by the admittivity at the Larmor frequency [9]. It does not require current injection and provides admittivity images at the Larmor frequency. Currently available practical MREPT methods [11] produce absolute admittivity images within local homogeneous regions where their admittivity values are constant. For a more general inhomogeneous admittivity distribution inside the human body, it is not clear how much error occurs in a reconstructed MREPT image. Most MREPT methods use only the positive rotating magnetic field since the negative rotating magnetic field is not available at present. Recently, Zhang et al. [27] suggested a dual-exand , but this method is citation approach to measure not easily applicable to most clinical MRI scanners. Since the quality of admittivity images in MREPT depends directly on the quality of measured B1 maps, a more thorough investigation is needed to understand effects of nonuniform admittivity values . and nonzero In this paper, we provide a rigorous mathematical error analysis of the MREPT admittivity image reconstruction problem. We will explicitly express the reconstruction error in MREPT and provide quantitative error analyses using numerical simulations and phantom experiments. Highlighting the difference between MREPT and MREIT, we will explain their pros and cons in terms of image reconstruction errors. Understanding the fundamental mechanism of image reconstruction errors in MREPT in comparison with MREIT, a novel MREPT admittivity image reconstruction technique is suggested to reduce the errors.
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magnetic field induced by an externally injected low-frequency current. B. MREPT Admittivity Image Reconstruction In MREPT, both terms in the right side of (3) should be dealt with to produce images of both and at the Larmor frequency. The RF magnetic field of the MRI scanner is denoted as and assume . Note that and . Most present MREPT image reconstruction methods ignore in (3) assuming that in the term a local region. It can safely be assumed that everywhere , the inside the human body. From the assumption of identity (3) becomes (4) Since each component of satisfies
satisfies the above identity,
(5) This provides the following direct reconstruction formula: (6) Indeed, if
, then
(7) II. MR-BASED ADMITTIVITY IMAGE RECONSTRUCTION PROBLEM
Assuming that the admittivity is constant within a disk centered at , Katscher et al. [11] proposed the following reconstruction formula:
A. Relation Between Admittivity and Magnetic Field From time-harmonic Maxwell equations
(8) (1) where is the electric field and Taking the curl operation to
is the admittivity.
(2) Since lead to the following relation:
, (1) and (2)
is the boundary of is the line element, and where is the surface element. The formula (8) is an implementation of a local average of (6). C. Error in MREPT Image Due to Assumption of Assuming a general inhomogeneous admittivity distribution inside the human body where and letting be an approximation of the inhomogeneous admittivity as (9)
(3) This relation between the admittivity and the magnetic field provides the physical basis for MR-based admittivity image reconstruction problems including MREPT and MREIT. In MREPT, is a magnetic field at the Larmor frequency induced by an RF coil whereas, in MREIT, is a low-frequency
of (7) under the condition that Note that . The reconstruction by using will produce a negligible amount of error if (10)
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Otherwise, there must occur a reconstruction error
as (11)
where depends upon the quantity in the left side of (10). To quantify the error , use the following identity by Nachman et al. [14]: (12) which can be derived from (3) and . Noting that
(13) it follows from (12) that
Observation 2.1: , then (no reconstruction error), i) If . Hence, the data is sufficient that is, for the accurate admittivity reconstruction by (9) when . and ii) If , then . Hence, the first term in the right side of (3) should be utilized to properly deal with the error . The proof of this observation is provided in the Appendix. Remark 2.2: According to the observation 2.1, the recon. The identity struction formula (9) cannot probe the contrast (8) appears to be approximately true for a general inhomogesufficiently small since can neous by taking the area be viewed as approximately a constant in a small local region. is not small, the However, if the magnitude of . identity (8) does not hold regardless of the size of Remark 2.3: From (12), Nachman et al. [14] derived the following exact formula of the admittivity: (18)
(14) where
However, this reconstruction formula (18) is very weak against measurement noise from an MRI scanner. The magnitude of can be small since and can be almost orthogonal in some regions. D. MREIT Conductivity Image Reconstruction Here, the MREIT conductivity image reconstruction problem in the context of the governing (3) will be briefly reviewed since and in we will compare them later. Since MREIT, the relation (3) can be simplified as
(15) (19) Note that we assume to
temporarily. Equation (14) leads Extracting the -component of (19) (20) (16)
For practical birdcage coils, the first two terms on the right side . Howof (15) is close to zero and can be ignored since ever, the third term still needs to be considered. Even though is negligible the z-component of , and hence , can be big enough to make the third term significant in (16). These will occur at nonconstant boundaries especially where the electric is significant, providing a rough guideline to the error’s field is small, an severity. In cases where the z-component of explicit expression for the error can be found using only and terms as
(17) From the above derivations, the followings can be observed.
and subject to two Assuming that two data sets of linearly independent injection currents and , respectively, have been obtained, the two linearly independent relations of the and can be formulated transversal contrast of with as
(21) and for in (21) are nonlinear functions Since algorithm [20] adopts an of the conductivity , the harmonic iterative approach to recover an absolute image of . A singlealgorithm produces a step implementation of the harmonic scaled conductivity image providing only contrast information. III. NUMERICAL SIMULATIONS Using an FDTD simulation software (REMCOM XFDTD ver. 7.1), we constructed models shown in Fig. 1. We placed a
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maximum conductivity decreases or if there is more change in conductivity. The normalized and absolute error in the permittivity increases if the maximum conductivity increases or if there is more change in conductivity as shown. Possible field explanations of this could include the decrease in value as the conductivity penetration and increase of is increased. In reality, the error would depend on the model used, its structure, spatial distribution, orientation, and so on. The illustration shows that even with a small spatial variation, can the resulting error due to ignoring be significantly large. Note that the error map will also depend and . In this on the angle between the vectors simulation, since the conductivity and permittivity is perfectly in the center of the symmetric for x-axis and y-axis, numerical phantom and thus the error term is zero here. In another set of simulations, the effects of discrete boundaries were examined. Here, we simulated both MREPT and MREIT image reconstructions using the model in Fig. 2(a) and (g), respectively. The intention was to confirm that MREPT can lead to incorrect quantification near boundaries with conis ignored. On the ductivity contrast since the term other hand, MREIT should shows less boundary effects since it . We used the single-step harmonic makes use of the term algorithm to produce scaled conductivity images in MREIT. Fig. 2 shows results for MREPT (a)–(f) and MREIT (g)–(l). Fig. 2(b) shows the reconstructed conductivity image using (9) while the error term is shown in (c). Fig. 2(d) shows (22) Fig. 1. MREPT simulation with spatially varying conductivity. (a) Cylindrical phantom with radially symmetric conductivity material placed inside a circularly polarized RF coil. (b), (c) Axial view of the phantom and the assumed spatial profile of the conductivity used for the simulations. Simulation results for different and showing (d) conductivity and (e) permittivity. (First row: true value; second row: estimated value; third row: absolute error; fourth row: normalized error, and maximum and rms of the normalized error values).
radially symmetric cylindrical object within a circularly polarized RF coil operating at 3T field strength. We assigned spatially varying conductivity distributions with various combinations of and . Setting the relative permittivity as 80, we computed maps of the true , estimated , error , and . normalized error Fig. 1 shows the MREPT simulation results for conductivity combinations of . The first three combinations assumed linearly varying conductivity with values while the later three combinations assumed same ratio with the same . Conductivity and different permittivity images are provided. The first three rows in Fig. 1(d)–(f) show the true values, estimated values, error values. The last row shows the normalized error term along with its maximum and root means square (rms) value. There occurred substantial errors in estimating the true , reaching values of over 50% for some cases. In general, the absolute error in conductivity increases as the conductivity increases. The normalized error in the conductivity increases if the
Note that the image of (d) appears to be very similar to the image of (c). The horizontal and vertical conductivity line profiles are given in (e) and (f), respectively. The green line represents the actual conductivity while the blue line represents estimated conductivity using (9). For MREPT, note that there occurred large deviations from the actual values at the boundaries. Away from the boundaries, MREPT provided accurate absolute values. Corresponding results for MREIT are shown in (j), (k), and (l). Though MREIT does not suffer from such boundary effects, it does not provide absolute conductivity values unless an iterative process is adopted. IV. MREPT PHANTOM EXPERIMENTS Fig. 3 shows a cylindrical phantom including three cylindrical anomalies with different conductivity values. The background was a mixture of 3 L distilled water and 75 g agar powder. For the three anomalies of agarose gel, we used 1.5%, 3%, and 4.5% NaCl solutions with 2.5% agar powder to produce three different conductivity values. We placed each agarose gel within a thin plastic bottle to prevent conductivity changes by ion diffusion. All three agarose gels and the background had 0.2% CuSO to reduce their T1 values. , we employed a To measure the amplitude and phase of B1 mapping scheme based on the double angle method (DAM) [23]. We acquired a set of spin echo images using 60 –180 flip angles and 120 –180 flip angles. We obtained the magnitude of by computing where and cor-
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Fig. 3. MREPT experiment to evaluate reconstruction errors in local regions where = = 0: (a) spin echo magnitude image, (b) B phase image, (c) B magnitude image, (d) photo of phantom, (e) reconstructed conductivity image with the actual phantom size marked (in yellow), and (f) conductivity profile along the dotted line marked in (e) with the actual phantom size marked (in red).
r
Fig. 2. MREPT (a)–(f) and MREIT (g)–(l) conductivity simulation showing boundary effects. (a), (g) Phantom model and its assumed conductivity for MREPT and MREIT, respectively. (b)–(f) MREPT simulation results: (b) reconstructed , (c) , (d) the real part of ([( )=( ) ( )] )=(! H ), (e), (f) horizontal and vertical line profile (green: true conductivity; blue: estimated conductivity). (h)–(l) MREIT simulation results: (h), (i) B map for horizontal and vertical current injection, (j) reconstructed using the standard Harmonic B algorithm, (k)–(l) horizontal and vertical line profile.
r
2 r2H
0
respond to the image magnitudes acquired from the 60 –180 and 120 –180 flip angles acquisitions, respectively. The phase was obtained from the DAM acquisition data by reof trieving the one-half value of the phase [24]. Imaging paramms, FOV eters were as follows: mm resolution mm , slice thickness mm, eight slices, and quadrature transmit/receive coil. All experiments were performed on a 3T scanner (Siemens Medical Solutions, Erlangen, Germany). Fig. 3 shows the results of the MREPT experiment. From the acquired B1 phase and magnitude maps, we reconstructed the conductivity image using the formula (9), which correctly recovered conductivity values inside the three homogeneous anomalies. To deal with noise, we used a smooth mollifier which was a Gaussian filter with kernel size 5 and standard deviation 1.0. Specifically, the filter was defined as a normalized function of the following: (23)
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Fig. 4. MREPT reconstruction near boundary. (a) Phantom constructed with a cylindrical material with different conductivity placed inside a background material. (b) MREPT reconstruction using (9) without any filtering. (c) MREPT reconstruction using (9) with a smooth mollifier. The smooth mollifier was a Gaussian kernel with kernel width 3 and standard deviation 1.0.
where the range of covers the kernel size and represents the standard deviation. The actual reconstruction therefore used the following formula: (24) where represents the convolution operator. In the MR magnitude image, the dark circular rings are the thin plastic bottle which enclosed the three anomalies. Around the boundary of each anomaly where the assumption of the local homogeneity is not satisfied, we can observe dark circular rings of which width is much thicker than that shown in the MR magnitude image. We examined the conductivity profile corresponding to the dashed line marked on the conductivity image. The profile clearly shows that reconstruction errors are significantly larger around the boundaries of the anomalies where the local homogeneity assumption fails. The size of this error depends on the degree of the local inhomogeneity. Though the plastic bottles were useful to preserve constant conductivity contrasts, they could have affected the conductivity estimates near the boundary due to their finite thickness. We, therefore, constructed another phantom where we placed an agarose gel in the background material without using the plastic bottle. Fig. 4 shows MREPT conductivity errors near the
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and unwrapped agarose objects, the reconstructed MREPT images showed erroneous regions around the boundaries of the objects as explained in the previous section regardless of filtering. However, MREIT shows uniform boundaries for both . The phantoms since they use the term slight bright boundaries in (c) are most likely due to diffusion effects that occurred after the phantom was built. VI. DISCUSSION
Fig. 5. Conductivity imaging experiment using MREIT and MREPT. (a) Magnitude image of the phantom and conductivity values of three local regions. (b), (c) Reconstructed low-frequency conductivity images from MREIT. (d), (e) Reconstructed high-frequency conductivity images from MREPT. In (b) and (d), both agarose anomalies were wrapped by thin insulating films whereas they were not wrapped in (c) and (e).
boundary of the cylindrical agar object. We used (9) without any filtering to reconstruct the conductivity image in (b). We can observe severe reconstruction errors at the outermost boundary of the phantom itself as well as the boundary of the agar object inside the phantom. In (c), we used a Gaussian kernel of size 3 and standard deviation 1.0 prior to the image reconstruction. This could increase SNR but also increased the size of the dark rims which correspond to the regions of erroneous conductivity estimates. V. MREIT AND MREPT PHANTOM EXPERIMENTS We conducted both MREIT and MREPT imaging experiments using two different phantoms. We prepared two cylindrical agarose gel objects with 1.14 and 2.79 S/m conductivity values. We wrapped those agarose gel objects with very thin plastic films and placed them inside the first phantom. In the second phantom with the same size and shape, we placed the agarose gel objects without using the thin plastic wrap. We filled the backgrounds of both phantoms with a saline of 0.12 S/m conductivity. We expected that the wrapped agarose gel objects should appear as insulators in an MREIT image since injected low-frequency current cannot penetrate the insulating thin films. However, the same wrapped agarose objects should show conductivity values in an MREPT image since the thin films are transparent to the electromagnetic wave at the Larmor frequency of 128 MHz. Both of the unwrapped agarose objects should show their conductivity values in both MREIT and MREPT images. For the MREIT experiment, we field mapping sequence for used a fast spin echo based data collection. The injected current amplitude was 3 mA with 85 ms pulse width. The MREPT experiment was performed separately using the method described above on the same 3T scanner. Fig. 5 shows the obtained results. As expected, the wrapped agarose objects appeared as insulators in the low-frequency conductivity image from the MREIT experiment. On the other hand, the image from the MREPT experiment showed their conductivity values without being affected by the thin insulating films. Both MREIT and MREPT recovered conductivity contrasts of the unwrapped agarose objects. For both wrapped
According to (20) and (5), there exist distinct differences between MREIT and MREPT in the use of term in (3). As a result, MREPT fails to probe , whereas MREIT method uses this term and results in an increased performance in regions where conductivity changes. On the other hand, MREPT accurately recovers absolute conductivity values within locally homogeneous regions whereas MREIT requires a cumbersome iterative process to recover absolute conductivity values. Experimental results of MREIT and MREPT were presented to highlight these distinct features of both methods in probing the same object in terms of its lowand high-frequency conductivity distributions, respectively. In MREPT, a partial information of the magnetic field influenced by the admittivity is utilized to reconstruct images of the admittivity . We showed cannot probe . that the formula (6) using only Providing rigorous error analysis, the error was also quantified by performing numerical simulations and phantom experiments. The reconstruction error is closely related to [see Fig. 2(c) and (d)]. VII. SUGGESTION FOR FUTURE STUDIES For future studies, a new reconstruction formula for MREPT is suggested which will be potentially useful to recover . Note that the term in (3) should be utilized to properly deal with a general inhomogeneous adas in mittivity distribution. To focus on perceiving MREIT reconstruction based on (20), the influence of the term in (3) should be eliminated. One can eliminate in (3) by multiplying it by to get (25) The use of the basic property of leads to (26) where (27) From (25) and (27), the admittivity contrast the relation
satisfies (28)
We suggest future studies to combine (9) with (28) for better image reconstructions of a general inhomogeneous admittivity distribution. As in (20) in MREIT, the relation in (28) enables
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us to probe the change of in the direction . If the vector is real-valued, the MREIT technique can be applicable field . Unfortunately, the vector field is comto compute plex-valued, and one needs to interpret the unusual complex diin the real Euclidian space rectional derivative , which will be the main topic of our future study. Equation (18) eliminates the first term in the right side of (3), while (28) eliminates the second term in the right side of (3). Hence, (28) could have an advantage of visualizing the contrast . It can be expected that (28) provides an adjunct tool to the method in [11] for dealing with the fundamental drawback . of visualizing Remark 7.1: For reader’s convenience, a way of using (28) is a real vector field and under unrealistic assumptions that is explained. Let and be points in the imaging object and let be a parameterization of an arc joining and such that , and . Then
. In methods use it as a major term to perceive order to improve the image quality in MREPT for a general inhomogeneous subject, properly combining (9) with (28) has been suggested. Since MREPT and MREIT are complementary providing admittivity information at high and low frequency, respectively, we plan to conduct dual-frequency admittivity imaging by performing simultaneous MREPT and MREIT experiments.
APPENDIX A PROOF OF OBSERVATION 2.1 , then from If from (17). (7) and hence under the assumption of (ii). To derive Next, prove a contradiction, suppose . From (16) (30) From (13), (30) and
(29) This provides a way of perceiving the difference from via (28). Hence, if the value of is known, so is . However, in reality, and are complex and one cannot directly use the method like (29). VIII. CONCLUSION The image reconstruction error of MREPT resulting from the assumption of the local homogeneity has been analyzed. We validated the error analysis by performing the MREPT phantom simulations and experiments. Results show that the algorithm (9) accurately recovers conductivity values within locally homogeneous regions while serious reconstruction errors occur where the conductivity changes. Experimental results of the conductivity profile in Fig. 3(f) clearly reveals large errors of negative conductivity values at the boundaries of the three agarose gel . One can also observe that the recovobjects where ered conductivity values of the three agarose objects match very well with the true values when local homogeneity is guaranteed. This analysis is important when considering in vivo applications of MREPT such as tumor imaging since tumors in general have a very heterogeneous compound and local homogeneity cannot be maintained. According to the observation 2.1 and numerical simulations, the reconstruction error in MREPT is fundamental when the is ratio large. MREIT and MREPT images in Fig. 5 clearly illustrates distinct properties of the two methods as expected. MREIT is whereas MREPT capable of visualizing the contrast can not. On the other hand, MREPT is advantageous over MREIT in recovering an absolute value of inside a locally homogeneous region. Currently available MREPT methods whereas MREIT neglect
and (31) Hence, it follows from (3) and (31) that:
(32) This contradict the assumption of (ii), and, therefore,
.
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[11] U. Katscher, T. Voigt, C. Findeklee, P. Vernickel, K. Nehrke, and O. Dossel, “Determination of electrical conductivity and local SAR via B1 mapping,” IEEE Trans. Med. Imag., vol. 28, no. 9, pp. 1365–1374, Sep. 2009. [12] H. J. Kim, Y. T. Kim, A. S. Minhas, W. C. Jeong, E. J. Woo, J. K. Seo, and O. J. Kwon, “In vivo high-resolution conductivity imaging of the human leg using MREIT: The first human experiment,” IEEE Trans. Med. Imag., vol. 28, no. 11, pp. 1681–1687, Nov. 2009. [13] H. J. Kim, B. I. Lee, Y. Cho, Y. T. Kim, B. T. Kang, H. M. Park, S. L. Lee, J. K. Seo, and E. J. Woo, “Conductivity imaging of canine brain using a 3 T MREIT system: Postmortem experiments,” Physiol. Meas., vol. 28, pp. 1341–1353, 2007. [14] A. Nachman, D. Wang, W. Ma, and M. Joy, “A local formula for inhomogeneous complex conductivity as a function of the RF magnetic field,” in ISMRM 15th Sci. Meeting Exhibit., Berlin, Germany, 2007 [Online]. Available: http://www.ismrm.org/07/Unsolved.htm [15] P. Metherall, D. C. Barber, R. H. Smallwood, and B. H. Brown, “Threedimensional electrical impedance tomography,” Nature, vol. 380, pp. 509–512, 1996. [16] A. S. Minhas, W. C. Jeong, Y. T. Kim, H. J. Kim, T. H. Lee, and E. J. Woo, “MREIT of postmortem swine legs using carbon-hydrogel electrodes,” J. Biomed. Eng. Res., vol. 29, pp. 436–442, 2008. [17] P. F. van de Moortele, C. Akgun, G. Adriany, S. Moeller, J. Ritter, C. M. Collins, M. B. Smith, J. T. Vaughan, and K. Ugurbil, “B(1) destructive interferences and spatial phase patterns at 7 T with a head transceiver array coil,” Magn. Reson. Med., vol. 54, pp. 1503–1518, 2005. [18] S. H. Oh, B. I. Lee, E. J. Woo, S. Y. Lee, M. H. Cho, O. Kwon, and J. K. Seo, “Conductivity and current density image reconstruction using harmonic B algorithm in magnetic resonance electrical impedance tomography,” Phys. Med. Biol., vol. 48, pp. 3101–3016, 2003.
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[19] S. H. Oh, B. I. Lee, E. J. Woo, S. Y. Lee, T. S. Kim, O. Kwon, and J. K. Seo, “Electrical conductivity images of biological tissue phantoms in MREIT,” Physiol. Meas., vol. 26, pp. S279–S288, 2005. [20] J. K. Seo, J. R. Yoon, E. J. Woo, and O. Kwon, “Reconstruction of conductivity and current density images using only one component of magnetic field measurements,” IEEE Trans. Biomed. Eng., vol. 50, no. 9, pp. 1121–1124, Sep. 2003. [21] G. C. Scott, M. L. G. Joy, R. L. Armstrong, and R. M. Henkelman, “Measurement of nonuniform current density by magnetic resonance,” IEEE Trans. Med. Imag., vol. 10, no. 3, pp. 362–374, Sep. 1991. [22] G. C. Scott, M. L. G. Joy, R. L. Armstrong, and R. M. Henkelman, “Sensitivity of magnetic-resonance current density imaging,” J. Mag. Res., vol. 97, pp. 235–254, 1992. [23] R. Stollberger and P. Wach, “Imaging of the active B1 field in vivo,” Magn. Reson. Med., vol. 35, pp. 246–251, 1996. [24] H. Wen, “Noninvasive quantitative mapping of conductivity and dielectric distributions using RF wave propagation effects in high-field MRI,” in Proc. SPIE, 2003, vol. 5030, pp. 471–477. [25] E. J. Woo, S. Y. Lee, and C. W. Mun, “Impedance tomography using internal current density distribution measured by nuclear magnetic resonance,” in Proc. SPIE, 1994, vol. 2299, pp. 377–385. [26] N. Zhang, “Electrical impedance tomography based on current density imaging,” M.S. thesis, Dept. Elec. Eng., Univ. Toronto, Toronto, Canada, 1992. [27] X. Zhang, S. Zhu, and B. He, “Imaging electric properties of biological tissues by RF field mapping in MRI,” IEEE Trans. Med. Imag., vol. 29, no. 2, pp. 474–481, Feb. 2010. [28] J. Wang, M. Qiu, Q. X. Yang, M. B. Smith, and R. T. Constable, “Measurement and correction of transmitter and receiver induced nonuniformities in vivo,” Magn. Reson. Med., vol. 53, pp. 408–417, 2005.
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Error Analysis of Nonconstant Admittivity for MR-Based Electric Property Imaging Jin Keun Seo, Min-Oh Kim, Joonsung Lee, Narae Choi, Eung Je Woo, Member, IEEE, Hyung Joong Kim, Oh In Kwon, and Dong-Hyun Kim*, Member, IEEE
Abstract—Magnetic resonance electrical property tomography (MREPT) is a new imaging modality to visualize a distribution inside the human body where of admittivity and denote electrical conductivity and permittivity, respectively. Using B1 maps acquired by an magnetic resonance imaging and at the scanner, it produces cross-sectional images of Larmor frequency. Since current MREPT methods rely on an assumption of a locally homogeneous admittivity, there occurs a reconstruction error where this assumption fails. Rigorously analyzing the reconstruction error in MREPT, we showed that the error is fundamental and may cause technical difficulties in interpreting MREPT images of a general inhomogeneous object. We performed numerical simulations and phantom experiments to quantitatively support the error analysis. We compared the MREPT image reconstruction problem with that of magnetic resonance electrical impedance tomography (MREIT) to highlight distinct features of both methods to probe the same object in terms of its high- and low-frequency conductivity distributions, respectively. MREPT images showed large errors along boundaries where admittivity values changed whereas MREIT images showed no such boundary effects. Noting that MREIT makes use of the term neglected in MREPT, a novel MREPT admittivity image reconstruction method is proposed to deal with the boundary effects, which requires further investigation on the complex directional derivative in the real Euclidian space 3 .
= +
Index Terms—B1 map, conductivity, magnetic resonance electrical impedance tomography (MREIT), magnetic resonance electrical property tomography (MREPT), permittivity.
I. INTRODUCTION
T
OMOGRAPHIC imaging of electrical conductivity and permittivity distributions inside the human body has been an important research topic since they provide diagnostic information related with physiological and pathological conditions Manuscript received May 20, 2011; revised July 26, 2011, August 23, 2011; accepted September 26, 2011. Date of publication October 10, 2011; date of current version February 03, 2012. The work of J. K. Seo was supported by the WCU program through MEST/NRF (R31-2008-000-10049-0). The work of E. J. Woo was supported by the NRF grant funded by the Korea government (MEST) (20100018275). The work of D. H. Kim was supported by the KOSEF grant funded by the Korea government (MEST) (2010-0016421). Asterisk indicates corresponding author. J. K. Seo is with the Department of Computational Science and Engineering, Yonsei University, Seoul 120-749, South Korea (e-mail: [email protected]). M.-O. Kim and J. Lee, N. Choi are with the Department of Electrical and Electronic Engineering, Yonsei University, 120-749 Seoul, South Korea (e-mail: [email protected]). E. J. Woo and H. J. Kim are with the Department of Biomedical Engineering, Kyung Hee University, 446–701 Yongin, Gyeonggi, South Korea. O. I. Kwon is with the Department of Mathematics, Konkuk University, Seoul 143-701, South Korea. *D.-H. Kim is with the Department of Electrical and Electronic Engineering, Yonsei University, Seoul 120-749, South Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TMI.2011.2171000
of tissues and organs [5]–[8]. Throughout this paper, the conand permittivity are assumed to be ductivity isotropic and depend on the angular frequency and the po. The admittivity is denoted as sition . The human body is assumed to be homogeneous in terms of the magnetic permeability with the value of . During the last three decades, there have been numerous studies on electrical impedance tomography (EIT), which aims to reconstruct cross-sectional images of an admittivity distribution inside the human body [3], [10], [15]. Using boundary current-voltage data sets subject to multiple externally injected currents, a solution of an ill-posed nonlinear inverse problem provides cross-sectional images of the admittivity distribution. The measured boundary data sets are, however, insensitive to a local perturbation of the admittivity away from measuring points whereas they are sensitive to errors in the boundary geometry and electrode positions. Static EIT imaging to recover absolute admittivity values has not been successful so far. Though linearized difference EIT imaging methods are finding some clinical applications, the spatial resolution is relatively poor compared with other imaging modalities. For static admittivity image reconstructions, magnetic resonance electrical impedance tomography (MREIT) [2], [25], [26] and magnetic resonance electrical property tomography (MREPT) [9], [11] have been proposed to use internal data of induced magnetic fields. In MREIT, a low-frequency current below a few kilohertz is injected to induce a magnetic field , whose -component is measured from MR phase images using a magnetic resonance imaging (MRI) scanner with its main field in the -direction. The current-injection MRI technique to measure the induced data in a form of an image was originally proposed for magnetic resonance current density imaging (MRCDI) [21], [22]. Since MREIT relies on the magnetic field induced by low-frequency current, it can produce images of the conductivity distribution only at the low frequency. For MREIT conductivity image reconstructions, one may adopt the harmonic algorithm [20], which utilizes two data sets of and subject to two linearly independent injection currents and , respectively. Since the induced magnetic fields of and are influenced by the conductivity , the harmonic algorithm recovers based on the relation between the induced magnetic field and the conductivity. Numerical simulations and imaging experiments of phantoms, animals, and human subjects showed that MREIT can produce low-frequency conductivity images with a pixel size of about one millimeter using imaging currents of a few milliamperes [12], [13], [16], [18], [19]. For in vivo
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SEO et al.: ERROR ANALYSIS OF NONCONSTANT ADMITTIVITY FOR MR-BASED ELECTRIC PROPERTY IMAGING
human imaging, the sensitivity of the MR-based measurement technique needs to be enhanced to reduce the amount of injection current down to a level where internal current density distributions do not induce nerve stimulation. maps [1], [4], [17], [23], [28] that are inMREPT utilizes fluenced by the admittivity at the Larmor frequency [9]. It does not require current injection and provides admittivity images at the Larmor frequency. Currently available practical MREPT methods [11] produce absolute admittivity images within local homogeneous regions where their admittivity values are constant. For a more general inhomogeneous admittivity distribution inside the human body, it is not clear how much error occurs in a reconstructed MREPT image. Most MREPT methods use only the positive rotating magnetic field since the negative rotating magnetic field is not available at present. Recently, Zhang et al. [27] suggested a dual-exand , but this method is citation approach to measure not easily applicable to most clinical MRI scanners. Since the quality of admittivity images in MREPT depends directly on the quality of measured B1 maps, a more thorough investigation is needed to understand effects of nonuniform admittivity values . and nonzero In this paper, we provide a rigorous mathematical error analysis of the MREPT admittivity image reconstruction problem. We will explicitly express the reconstruction error in MREPT and provide quantitative error analyses using numerical simulations and phantom experiments. Highlighting the difference between MREPT and MREIT, we will explain their pros and cons in terms of image reconstruction errors. Understanding the fundamental mechanism of image reconstruction errors in MREPT in comparison with MREIT, a novel MREPT admittivity image reconstruction technique is suggested to reduce the errors.
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magnetic field induced by an externally injected low-frequency current. B. MREPT Admittivity Image Reconstruction In MREPT, both terms in the right side of (3) should be dealt with to produce images of both and at the Larmor frequency. The RF magnetic field of the MRI scanner is denoted as and assume . Note that and . Most present MREPT image reconstruction methods ignore in (3) assuming that in the term a local region. It can safely be assumed that everywhere , the inside the human body. From the assumption of identity (3) becomes (4) Since each component of satisfies
satisfies the above identity,
(5) This provides the following direct reconstruction formula: (6) Indeed, if
, then
(7) II. MR-BASED ADMITTIVITY IMAGE RECONSTRUCTION PROBLEM
Assuming that the admittivity is constant within a disk centered at , Katscher et al. [11] proposed the following reconstruction formula:
A. Relation Between Admittivity and Magnetic Field From time-harmonic Maxwell equations
(8) (1) where is the electric field and Taking the curl operation to
is the admittivity.
(2) Since lead to the following relation:
, (1) and (2)
is the boundary of is the line element, and where is the surface element. The formula (8) is an implementation of a local average of (6). C. Error in MREPT Image Due to Assumption of Assuming a general inhomogeneous admittivity distribution inside the human body where and letting be an approximation of the inhomogeneous admittivity as (9)
(3) This relation between the admittivity and the magnetic field provides the physical basis for MR-based admittivity image reconstruction problems including MREPT and MREIT. In MREPT, is a magnetic field at the Larmor frequency induced by an RF coil whereas, in MREIT, is a low-frequency
of (7) under the condition that Note that . The reconstruction by using will produce a negligible amount of error if (10)
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Otherwise, there must occur a reconstruction error
as (11)
where depends upon the quantity in the left side of (10). To quantify the error , use the following identity by Nachman et al. [14]: (12) which can be derived from (3) and . Noting that
(13) it follows from (12) that
Observation 2.1: , then (no reconstruction error), i) If . Hence, the data is sufficient that is, for the accurate admittivity reconstruction by (9) when . and ii) If , then . Hence, the first term in the right side of (3) should be utilized to properly deal with the error . The proof of this observation is provided in the Appendix. Remark 2.2: According to the observation 2.1, the recon. The identity struction formula (9) cannot probe the contrast (8) appears to be approximately true for a general inhomogesufficiently small since can neous by taking the area be viewed as approximately a constant in a small local region. is not small, the However, if the magnitude of . identity (8) does not hold regardless of the size of Remark 2.3: From (12), Nachman et al. [14] derived the following exact formula of the admittivity: (18)
(14) where
However, this reconstruction formula (18) is very weak against measurement noise from an MRI scanner. The magnitude of can be small since and can be almost orthogonal in some regions. D. MREIT Conductivity Image Reconstruction Here, the MREIT conductivity image reconstruction problem in the context of the governing (3) will be briefly reviewed since and in we will compare them later. Since MREIT, the relation (3) can be simplified as
(15) (19) Note that we assume to
temporarily. Equation (14) leads Extracting the -component of (19) (20) (16)
For practical birdcage coils, the first two terms on the right side . Howof (15) is close to zero and can be ignored since ever, the third term still needs to be considered. Even though is negligible the z-component of , and hence , can be big enough to make the third term significant in (16). These will occur at nonconstant boundaries especially where the electric is significant, providing a rough guideline to the error’s field is small, an severity. In cases where the z-component of explicit expression for the error can be found using only and terms as
(17) From the above derivations, the followings can be observed.
and subject to two Assuming that two data sets of linearly independent injection currents and , respectively, have been obtained, the two linearly independent relations of the and can be formulated transversal contrast of with as
(21) and for in (21) are nonlinear functions Since algorithm [20] adopts an of the conductivity , the harmonic iterative approach to recover an absolute image of . A singlealgorithm produces a step implementation of the harmonic scaled conductivity image providing only contrast information. III. NUMERICAL SIMULATIONS Using an FDTD simulation software (REMCOM XFDTD ver. 7.1), we constructed models shown in Fig. 1. We placed a
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maximum conductivity decreases or if there is more change in conductivity. The normalized and absolute error in the permittivity increases if the maximum conductivity increases or if there is more change in conductivity as shown. Possible field explanations of this could include the decrease in value as the conductivity penetration and increase of is increased. In reality, the error would depend on the model used, its structure, spatial distribution, orientation, and so on. The illustration shows that even with a small spatial variation, can the resulting error due to ignoring be significantly large. Note that the error map will also depend and . In this on the angle between the vectors simulation, since the conductivity and permittivity is perfectly in the center of the symmetric for x-axis and y-axis, numerical phantom and thus the error term is zero here. In another set of simulations, the effects of discrete boundaries were examined. Here, we simulated both MREPT and MREIT image reconstructions using the model in Fig. 2(a) and (g), respectively. The intention was to confirm that MREPT can lead to incorrect quantification near boundaries with conis ignored. On the ductivity contrast since the term other hand, MREIT should shows less boundary effects since it . We used the single-step harmonic makes use of the term algorithm to produce scaled conductivity images in MREIT. Fig. 2 shows results for MREPT (a)–(f) and MREIT (g)–(l). Fig. 2(b) shows the reconstructed conductivity image using (9) while the error term is shown in (c). Fig. 2(d) shows (22) Fig. 1. MREPT simulation with spatially varying conductivity. (a) Cylindrical phantom with radially symmetric conductivity material placed inside a circularly polarized RF coil. (b), (c) Axial view of the phantom and the assumed spatial profile of the conductivity used for the simulations. Simulation results for different and showing (d) conductivity and (e) permittivity. (First row: true value; second row: estimated value; third row: absolute error; fourth row: normalized error, and maximum and rms of the normalized error values).
radially symmetric cylindrical object within a circularly polarized RF coil operating at 3T field strength. We assigned spatially varying conductivity distributions with various combinations of and . Setting the relative permittivity as 80, we computed maps of the true , estimated , error , and . normalized error Fig. 1 shows the MREPT simulation results for conductivity combinations of . The first three combinations assumed linearly varying conductivity with values while the later three combinations assumed same ratio with the same . Conductivity and different permittivity images are provided. The first three rows in Fig. 1(d)–(f) show the true values, estimated values, error values. The last row shows the normalized error term along with its maximum and root means square (rms) value. There occurred substantial errors in estimating the true , reaching values of over 50% for some cases. In general, the absolute error in conductivity increases as the conductivity increases. The normalized error in the conductivity increases if the
Note that the image of (d) appears to be very similar to the image of (c). The horizontal and vertical conductivity line profiles are given in (e) and (f), respectively. The green line represents the actual conductivity while the blue line represents estimated conductivity using (9). For MREPT, note that there occurred large deviations from the actual values at the boundaries. Away from the boundaries, MREPT provided accurate absolute values. Corresponding results for MREIT are shown in (j), (k), and (l). Though MREIT does not suffer from such boundary effects, it does not provide absolute conductivity values unless an iterative process is adopted. IV. MREPT PHANTOM EXPERIMENTS Fig. 3 shows a cylindrical phantom including three cylindrical anomalies with different conductivity values. The background was a mixture of 3 L distilled water and 75 g agar powder. For the three anomalies of agarose gel, we used 1.5%, 3%, and 4.5% NaCl solutions with 2.5% agar powder to produce three different conductivity values. We placed each agarose gel within a thin plastic bottle to prevent conductivity changes by ion diffusion. All three agarose gels and the background had 0.2% CuSO to reduce their T1 values. , we employed a To measure the amplitude and phase of B1 mapping scheme based on the double angle method (DAM) [23]. We acquired a set of spin echo images using 60 –180 flip angles and 120 –180 flip angles. We obtained the magnitude of by computing where and cor-
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Fig. 3. MREPT experiment to evaluate reconstruction errors in local regions where = = 0: (a) spin echo magnitude image, (b) B phase image, (c) B magnitude image, (d) photo of phantom, (e) reconstructed conductivity image with the actual phantom size marked (in yellow), and (f) conductivity profile along the dotted line marked in (e) with the actual phantom size marked (in red).
r
Fig. 2. MREPT (a)–(f) and MREIT (g)–(l) conductivity simulation showing boundary effects. (a), (g) Phantom model and its assumed conductivity for MREPT and MREIT, respectively. (b)–(f) MREPT simulation results: (b) reconstructed , (c) , (d) the real part of ([( )=( ) ( )] )=(! H ), (e), (f) horizontal and vertical line profile (green: true conductivity; blue: estimated conductivity). (h)–(l) MREIT simulation results: (h), (i) B map for horizontal and vertical current injection, (j) reconstructed using the standard Harmonic B algorithm, (k)–(l) horizontal and vertical line profile.
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2 r2H
0
respond to the image magnitudes acquired from the 60 –180 and 120 –180 flip angles acquisitions, respectively. The phase was obtained from the DAM acquisition data by reof trieving the one-half value of the phase [24]. Imaging paramms, FOV eters were as follows: mm resolution mm , slice thickness mm, eight slices, and quadrature transmit/receive coil. All experiments were performed on a 3T scanner (Siemens Medical Solutions, Erlangen, Germany). Fig. 3 shows the results of the MREPT experiment. From the acquired B1 phase and magnitude maps, we reconstructed the conductivity image using the formula (9), which correctly recovered conductivity values inside the three homogeneous anomalies. To deal with noise, we used a smooth mollifier which was a Gaussian filter with kernel size 5 and standard deviation 1.0. Specifically, the filter was defined as a normalized function of the following: (23)
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Fig. 4. MREPT reconstruction near boundary. (a) Phantom constructed with a cylindrical material with different conductivity placed inside a background material. (b) MREPT reconstruction using (9) without any filtering. (c) MREPT reconstruction using (9) with a smooth mollifier. The smooth mollifier was a Gaussian kernel with kernel width 3 and standard deviation 1.0.
where the range of covers the kernel size and represents the standard deviation. The actual reconstruction therefore used the following formula: (24) where represents the convolution operator. In the MR magnitude image, the dark circular rings are the thin plastic bottle which enclosed the three anomalies. Around the boundary of each anomaly where the assumption of the local homogeneity is not satisfied, we can observe dark circular rings of which width is much thicker than that shown in the MR magnitude image. We examined the conductivity profile corresponding to the dashed line marked on the conductivity image. The profile clearly shows that reconstruction errors are significantly larger around the boundaries of the anomalies where the local homogeneity assumption fails. The size of this error depends on the degree of the local inhomogeneity. Though the plastic bottles were useful to preserve constant conductivity contrasts, they could have affected the conductivity estimates near the boundary due to their finite thickness. We, therefore, constructed another phantom where we placed an agarose gel in the background material without using the plastic bottle. Fig. 4 shows MREPT conductivity errors near the
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and unwrapped agarose objects, the reconstructed MREPT images showed erroneous regions around the boundaries of the objects as explained in the previous section regardless of filtering. However, MREIT shows uniform boundaries for both . The phantoms since they use the term slight bright boundaries in (c) are most likely due to diffusion effects that occurred after the phantom was built. VI. DISCUSSION
Fig. 5. Conductivity imaging experiment using MREIT and MREPT. (a) Magnitude image of the phantom and conductivity values of three local regions. (b), (c) Reconstructed low-frequency conductivity images from MREIT. (d), (e) Reconstructed high-frequency conductivity images from MREPT. In (b) and (d), both agarose anomalies were wrapped by thin insulating films whereas they were not wrapped in (c) and (e).
boundary of the cylindrical agar object. We used (9) without any filtering to reconstruct the conductivity image in (b). We can observe severe reconstruction errors at the outermost boundary of the phantom itself as well as the boundary of the agar object inside the phantom. In (c), we used a Gaussian kernel of size 3 and standard deviation 1.0 prior to the image reconstruction. This could increase SNR but also increased the size of the dark rims which correspond to the regions of erroneous conductivity estimates. V. MREIT AND MREPT PHANTOM EXPERIMENTS We conducted both MREIT and MREPT imaging experiments using two different phantoms. We prepared two cylindrical agarose gel objects with 1.14 and 2.79 S/m conductivity values. We wrapped those agarose gel objects with very thin plastic films and placed them inside the first phantom. In the second phantom with the same size and shape, we placed the agarose gel objects without using the thin plastic wrap. We filled the backgrounds of both phantoms with a saline of 0.12 S/m conductivity. We expected that the wrapped agarose gel objects should appear as insulators in an MREIT image since injected low-frequency current cannot penetrate the insulating thin films. However, the same wrapped agarose objects should show conductivity values in an MREPT image since the thin films are transparent to the electromagnetic wave at the Larmor frequency of 128 MHz. Both of the unwrapped agarose objects should show their conductivity values in both MREIT and MREPT images. For the MREIT experiment, we field mapping sequence for used a fast spin echo based data collection. The injected current amplitude was 3 mA with 85 ms pulse width. The MREPT experiment was performed separately using the method described above on the same 3T scanner. Fig. 5 shows the obtained results. As expected, the wrapped agarose objects appeared as insulators in the low-frequency conductivity image from the MREIT experiment. On the other hand, the image from the MREPT experiment showed their conductivity values without being affected by the thin insulating films. Both MREIT and MREPT recovered conductivity contrasts of the unwrapped agarose objects. For both wrapped
According to (20) and (5), there exist distinct differences between MREIT and MREPT in the use of term in (3). As a result, MREPT fails to probe , whereas MREIT method uses this term and results in an increased performance in regions where conductivity changes. On the other hand, MREPT accurately recovers absolute conductivity values within locally homogeneous regions whereas MREIT requires a cumbersome iterative process to recover absolute conductivity values. Experimental results of MREIT and MREPT were presented to highlight these distinct features of both methods in probing the same object in terms of its lowand high-frequency conductivity distributions, respectively. In MREPT, a partial information of the magnetic field influenced by the admittivity is utilized to reconstruct images of the admittivity . We showed cannot probe . that the formula (6) using only Providing rigorous error analysis, the error was also quantified by performing numerical simulations and phantom experiments. The reconstruction error is closely related to [see Fig. 2(c) and (d)]. VII. SUGGESTION FOR FUTURE STUDIES For future studies, a new reconstruction formula for MREPT is suggested which will be potentially useful to recover . Note that the term in (3) should be utilized to properly deal with a general inhomogeneous adas in mittivity distribution. To focus on perceiving MREIT reconstruction based on (20), the influence of the term in (3) should be eliminated. One can eliminate in (3) by multiplying it by to get (25) The use of the basic property of leads to (26) where (27) From (25) and (27), the admittivity contrast the relation
satisfies (28)
We suggest future studies to combine (9) with (28) for better image reconstructions of a general inhomogeneous admittivity distribution. As in (20) in MREIT, the relation in (28) enables
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us to probe the change of in the direction . If the vector is real-valued, the MREIT technique can be applicable field . Unfortunately, the vector field is comto compute plex-valued, and one needs to interpret the unusual complex diin the real Euclidian space rectional derivative , which will be the main topic of our future study. Equation (18) eliminates the first term in the right side of (3), while (28) eliminates the second term in the right side of (3). Hence, (28) could have an advantage of visualizing the contrast . It can be expected that (28) provides an adjunct tool to the method in [11] for dealing with the fundamental drawback . of visualizing Remark 7.1: For reader’s convenience, a way of using (28) is a real vector field and under unrealistic assumptions that is explained. Let and be points in the imaging object and let be a parameterization of an arc joining and such that , and . Then
. In methods use it as a major term to perceive order to improve the image quality in MREPT for a general inhomogeneous subject, properly combining (9) with (28) has been suggested. Since MREPT and MREIT are complementary providing admittivity information at high and low frequency, respectively, we plan to conduct dual-frequency admittivity imaging by performing simultaneous MREPT and MREIT experiments.
APPENDIX A PROOF OF OBSERVATION 2.1 , then from If from (17). (7) and hence under the assumption of (ii). To derive Next, prove a contradiction, suppose . From (16) (30) From (13), (30) and
(29) This provides a way of perceiving the difference from via (28). Hence, if the value of is known, so is . However, in reality, and are complex and one cannot directly use the method like (29). VIII. CONCLUSION The image reconstruction error of MREPT resulting from the assumption of the local homogeneity has been analyzed. We validated the error analysis by performing the MREPT phantom simulations and experiments. Results show that the algorithm (9) accurately recovers conductivity values within locally homogeneous regions while serious reconstruction errors occur where the conductivity changes. Experimental results of the conductivity profile in Fig. 3(f) clearly reveals large errors of negative conductivity values at the boundaries of the three agarose gel . One can also observe that the recovobjects where ered conductivity values of the three agarose objects match very well with the true values when local homogeneity is guaranteed. This analysis is important when considering in vivo applications of MREPT such as tumor imaging since tumors in general have a very heterogeneous compound and local homogeneity cannot be maintained. According to the observation 2.1 and numerical simulations, the reconstruction error in MREPT is fundamental when the is ratio large. MREIT and MREPT images in Fig. 5 clearly illustrates distinct properties of the two methods as expected. MREIT is whereas MREPT capable of visualizing the contrast can not. On the other hand, MREPT is advantageous over MREIT in recovering an absolute value of inside a locally homogeneous region. Currently available MREPT methods whereas MREIT neglect
and (31) Hence, it follows from (3) and (31) that:
(32) This contradict the assumption of (ii), and, therefore,
.
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SEO et al.: ERROR ANALYSIS OF NONCONSTANT ADMITTIVITY FOR MR-BASED ELECTRIC PROPERTY IMAGING
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